cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 37 results. Next

A334607 Number of Pythagorean triangles with perimeter A010814(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 2, 1, 2, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 5, 2, 2, 1, 1, 1, 2, 2, 2, 2, 1, 4, 1, 1, 1
Offset: 1

Views

Author

Wesley Ivan Hurt, May 08 2020

Keywords

Examples

			a(1) = 1; There is one integer-sided right triangle with perimeter A010814(1) = 12, [3,4,5].
a(2) = 1; There is one integer-sided right triangle with perimeter A010814(2) = 24, [6,8,10].

Links

Crossrefs

Cf. A005044, A010814.

A334678 Largest possible hypotenuse of a Pythagorean triangle with perimeter A010814(n).

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 26, 29, 30, 34, 37, 41, 40, 45, 50, 52, 53, 61, 58, 65, 65, 65, 65, 68, 75, 73, 82, 85, 80, 85, 85, 85, 89, 91, 90, 101, 100, 95, 97, 113, 111, 109, 122, 123, 115, 125, 125, 130, 130, 145, 130, 145, 136, 135, 143, 150, 149, 145, 145, 146, 164, 170, 155
Offset: 1

Views

Author

Wesley Ivan Hurt, May 08 2020

Keywords

Examples

			a(1) = 5; There is one integer-sided right triangle with perimeter A010814(1) = 12, [3,4,5] with hypotenuse 5.
a(2) = 10; There is one integer-sided right triangle with perimeter A010814(2) = 24, [6,8,10] with hypotenuse 10.

Links

Crossrefs

Cf. A005044, A010814, A334607.

A334717 Largest possible short leg length of a Pythagorean triangle with perimeter A010814(n).

Original entry on oeis.org

3, 6, 5, 9, 8, 12, 7, 15, 20, 18, 16, 21, 15, 24, 27, 14, 30, 28, 33, 40, 36, 25, 33, 39, 32, 42, 48, 45, 13, 48, 36, 40, 51, 39, 60, 54, 20, 28, 57, 65, 60, 63, 60, 66, 45, 69, 80, 44, 72, 75, 17, 66, 78, 64, 81, 88, 84, 51, 87, 100, 96, 90, 26, 93, 85, 84, 19, 96, 65, 49, 99
Offset: 1

Views

Author

Wesley Ivan Hurt, May 08 2020

Keywords

Examples

			a(1) = 3; There is one integer-sided right triangle with perimeter A010814(1) = 12, [3,4,5] with short leg length 3.
a(2) = 6; There is one integer-sided right triangle with perimeter A010814(2) = 24, [6,8,10] with short leg length 6.

Links

Crossrefs

Cf. A005044, A010814, A334607, A334678.

A380875 Indices of triangular numbers (A000217) which are also perimeters of integer-sided right triangles (A010814).

Original entry on oeis.org

8, 15, 20, 23, 24, 27, 32, 35, 39, 44, 47, 48, 51, 55, 56, 59, 60, 63, 64, 68, 71, 72, 75, 76, 79, 80, 84, 87, 91, 92, 95, 96, 99, 104, 111, 112, 115, 116, 119, 120, 123, 124, 128, 132, 135, 139, 140, 143, 144, 147, 152, 155, 159, 160, 164, 167, 168, 171, 175, 176, 179, 180, 183, 184, 187, 188
Offset: 1

Views

Author

M. F. Hasler, Mar 20 2025

Keywords

Comments

This is relevant for considering integer-sided right triangles whose sides are made of sticks of length 1, 2, 3, ..., n, using all those.
Sequence A382268 considers the special case where these sticks must be used in order.

Links

Crossrefs

Cf. A000217 (triangular numbers), A010814 (perimeters of right triangles), A382268 (subsequence of k for which a right triangle can be made with linked rods of length 1, ..., k).

Programs

A136000 a(n) = A010814(n) - 1.

Original entry on oeis.org

11, 23, 29, 35, 39, 47, 55, 59, 69, 71, 79, 83, 89, 95, 107, 111, 119, 125, 131, 139, 143, 149, 153, 155, 159, 167, 175, 179, 181, 191, 197, 199, 203, 207, 209, 215, 219, 223, 227, 233, 239, 251, 259, 263, 269, 275, 279, 285, 287, 299, 305, 307, 311, 319, 323
Offset: 1

Views

Author

Omar E. Pol, Dec 10 2007

Keywords

Comments

Numbers of the form P-1 in increasing order, where P is the sum of a Pythagorean triple. Also P is the perimeter of a Pythagorean triangle. The open triangle represent a triangle instrument and, in general, any musical instrument. Positive integers are musician numbers or dancer number A136002.

Examples

			a(1) = 11 because {3,4,5} is a Pythagorean triple and 3+4+5 = 12 is the sum of a Pythagorean triple and 11+1 = 12, then we can write 3+4+5 = 11+1.

Links

Crossrefs

Cf. A010814, A136001, A136002, A009096 (perimeters of Pythagorean triangles).

Programs

  • Mathematica
    q[n_] := OddQ[n] && Module[{d = Divisors[(n+1)/2]}, AnyTrue[Range[3, Length[d]], d[[#]] < 2 * d[[#-1]] &]]; Select[Range[350], q] (* Amiram Eldar, Oct 19 2024 *)

Extensions

Definition corrected by R. J. Mathar, Dec 12 2007
Extended by Ray Chandler, Dec 13 2008

A334753 Sum of the lengths of the short legs of all Pythagorean triangles with perimeter A010814(n).

Original entry on oeis.org

3, 6, 5, 9, 8, 12, 7, 25, 20, 18, 16, 33, 24, 24, 27, 14, 74, 28, 44, 40, 52, 25, 33, 39, 32, 87, 48, 93, 13, 48, 36, 40, 51, 39, 95, 54, 20, 28, 57, 65, 163, 155, 60, 88, 72, 69, 171, 44, 104, 125, 17, 66, 102, 64, 81, 143, 174, 51, 87, 100, 96, 258, 26, 93, 85, 84, 19, 96
Offset: 1

Views

Author

Wesley Ivan Hurt, May 10 2020

Keywords

Examples

			a(1) = 3; There is one integer-sided right triangle with perimeter A010814(1) = 12, [3,4,5] with short leg length 3.
a(2) = 6; There is one integer-sided right triangle with perimeter A010814(2) = 24, [6,8,10] with short leg length 6.

Links

Crossrefs

Cf. A005044, A010814, A334607, A334678, A334717.
Cf. A334755 (sum of longest legs), A334757 (sum of hypotenuses).

A334755 Sum of the lengths of the long legs of all Pythagorean triangles with perimeter A010814(n).

Original entry on oeis.org

4, 8, 12, 12, 15, 16, 24, 44, 21, 24, 30, 63, 76, 32, 36, 48, 133, 45, 104, 42, 111, 60, 56, 52, 60, 198, 55, 212, 84, 64, 77, 75, 68, 80, 147, 72, 99, 96, 76, 72, 378, 279, 91, 208, 228, 92, 309, 117, 222, 220, 144, 112, 247, 120, 108, 237, 396, 140, 116, 105, 110, 559
Offset: 1

Views

Author

Wesley Ivan Hurt, May 10 2020

Keywords

Examples

			a(1) = 4; There is one integer-sided right triangle with perimeter A010814(1) = 12, [3,4,5] with long leg 4.
a(2) = 8; There is one integer-sided right triangle with perimeter A010814(2) = 24, [6,8,10] with long leg 8.

Links

Crossrefs

Cf. A005044, A010814, A334607, A334678, A334717.
Cf. A334753 (sum of short legs), A334757 (sum of hypotenuses).

A334757 Sum of the lengths of the hypotenuses of all Pythagorean triangles with perimeter A010814(n).

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 51, 29, 30, 34, 72, 80, 40, 45, 50, 153, 53, 116, 58, 125, 65, 65, 65, 68, 219, 73, 235, 85, 80, 85, 85, 85, 89, 178, 90, 101, 100, 95, 97, 419, 322, 109, 232, 240, 115, 360, 125, 250, 255, 145, 130, 275, 136, 135, 280, 438, 149, 145, 145, 146, 623
Offset: 1

Views

Author

Wesley Ivan Hurt, May 10 2020

Keywords

Examples

			a(1) = 4; There is one integer-sided right triangle with perimeter A010814(1) = 12, [3,4,5] with hypotenuse 5.
a(2) = 8; There is one integer-sided right triangle with perimeter A010814(2) = 24, [6,8,10] with hypotenuse 10.

Links

Crossrefs

Cf. A005044, A010814, A334607, A334678, A334717.
Cf. A334753 (sum of short legs), A334755 (sum of long legs).

A332962 Sum of the areas of all Pythagorean triangles with perimeter A010814(n).

Original entry on oeis.org

6, 24, 30, 54, 60, 96, 84, 270, 210, 216, 240, 504, 450, 384, 486, 336, 1620, 630, 1056, 840, 1368, 750, 924, 1014, 960, 2772, 1320, 3150, 546, 1536, 1386, 1500, 1734, 1560, 3360, 1944, 990, 1344, 2166, 2340, 7320, 7056, 2730, 4224, 4050, 3174, 8400, 2574, 5472, 6750
Offset: 1

Views

Author

Wesley Ivan Hurt, May 10 2020

Keywords

Examples

			a(1) = 6; There is one Pythagorean triangle with perimeter A010814(1) = 12, [3,4,5] with area 3*4/2 = 6.
a(2) = 24; There is one Pythagorean triangle with perimeter A010814(2) = 24, [6,8,10] with area 6*8/2 = 24.

Links

Crossrefs

Cf. A010814, A334788.

A334788 Largest possible area of a Pythagorean triangle with perimeter A010814(n).

Original entry on oeis.org

6, 24, 30, 54, 60, 96, 84, 150, 210, 216, 240, 294, 270, 384, 486, 336, 600, 630, 726, 840, 864, 750, 924, 1014, 960, 1176, 1320, 1350, 546, 1536, 1386, 1500, 1734, 1560, 1890, 1944, 990, 1344, 2166, 2340, 2400, 2646, 2730, 2904, 2430, 3174, 3360, 2574, 3456, 3750, 1224
Offset: 1

Views

Author

Wesley Ivan Hurt, May 10 2020

Keywords

Examples

			a(1) = 6; There is one Pythagorean triangle with perimeter A010814(1) = 12, [3,4,5] with area 3*4/2 = 6.
a(2) = 24; There is one Pythagorean triangle with perimeter A010814(2) = 24, [6,8,10] with area 6*8/2 = 24.

Links

Crossrefs

Cf. A010814, A332962.
Showing 1-10 of 37 results. Next

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